Question

evaluate the determinant.\

$$\begin{vmatrix}3&8&2\\\\6&-1&-3\\\\-3&-3&1\\end{vmatrix}$$

Explanation:

Step1: Recall the formula for the determinant of a 3x3 matrix

$$\begin{vmatrix}a&b&c\\d&e&f\\g&h&i\end{vmatrix}=a(ei - fh)-b(di - fg)+c(dh - eg)$ For the given matrix $\begin{vmatrix}3&8&2\\6&-1&-3\\-3&-3&1\end{vmatrix}$$

, we have $a = 3$, $b = 8$, $c = 2$, $d = 6$, $e=-1$, $f = -3$, $g=-3$, $h=-3$, $i = 1$.

Step2: Calculate each part

First, calculate $ei - fh$: $(-1)(1)-(-3)(-3)=-1 - 9=-10$
Then, calculate $di - fg$: $6(1)-(-3)(-3)=6 - 9=-3$
Next, calculate $dh - eg$: $6(-3)-(-1)(-3)=-18 - 3=-21$

Step3: Substitute into the formula

$a(ei - fh)-b(di - fg)+c(dh - eg)=3(-10)-8(-3)+2(-21)$
$=-30 + 24-42$
$=-30+24=-6$; $-6-42=-48$

Answer:

-48